# (Probably difficult) inequality question for densities under different meaures

Given the measures $(G_0,G_1)$ and $(G^{'}_0,G^{'}_1)$ corresponding to the densities $(g_0,g_1)$ and $(g^{'}_0,g^{'}_1)$ the following inequality

$$G_0[g_1/g_0<t^{'}]\geq G^{'}_0[g_1/g_0<t^{'}]\geq G^{'}_1[g_1/g_0<t^{'}]\geq G_1[g_1/g_0<t^{'}]$$

is satisfied for all $t^{'}$. Lets assume that everything runs on the real numbers, i.e., $t^{'}\in\mathbb{R}$ and $(g_i,g_i^{'})\in\mathbb{R}$ for $i=0,1$. Furthermore $g_1/g_0$ is monotonically increasing.

I am also given the following pairs of densities $$q_0(x)=(1-\epsilon_0)g_0(x)\quad\quad\quad\quad\quad for\quad g_1(x)/g_0(x)<c_u$$ $$\quad=(1/c_u)(1-\epsilon_0)g_1(x)\quad\quad\quad\quad for \quad g_1(x)/g_0(x)\geq c_u$$

$$q_1(x)=(1-\epsilon_1)g_1(x)\quad\quad\quad\quad\quad for\quad g_1(x)/g_0(x)>c_l$$ $$\quad=c_l(1-\epsilon_1)g_0(x)\quad\quad\quad\quad\quad\quad for \quad g_1(x)/g_0(x)\leq c_l$$ ..................................................................................................................................................................... $$q^{'}_0(x)=(1-\epsilon_0)g^{'}_0(x)\quad\quad\quad\quad\quad for\quad g^{'}_1(x)/g^{'}_0(x)<c^{'}_u$$ $$\quad=(1/c^{'}_u)(1-\epsilon_0)g^{'}_1(x)\quad\quad\quad\quad for \quad g^{'}_1(x)/g^{'}_0(x)\geq c^{'}_u$$

$$q^{'}_1(x)=(1-\epsilon_1)g^{'}_1(x)\quad\quad\quad\quad\quad for\quad g^{'}_1(x)/g^{'}_0(x)>c^{'}_l$$ $$\quad=c^{'}_l(1-\epsilon_1)g^{'}_0(x)\quad\quad\quad\quad\quad\quad for \quad g^{'}_1(x)/g^{'}_0(x)\leq c^{'}_l$$

where $0<\epsilon_0,\epsilon_1<1$ and the constants $c_l, c_u, c^{'}_l,c^{'}_u$ satisfy

$$(1-\epsilon_0)\{G_0[g_1/g_0<c_u]+c^{-1}_u G_1[g_1/g_0\geq c_u]\}=1$$ $$(1-\epsilon_1)\{G_1[g_1/g_0>c_l]+c_l G_0[g_1/g_0\leq c_l]\}=1$$ $$(1-\epsilon_0)\{G_0^{'}[g^{'}_1/g^{'}_0<c^{'}_u]+c^{'\,-1}_u G_1[g^{'}_1/g^{'}_0\geq c_u]\}=1$$ $$(1-\epsilon_1)\{G_1^{'}[g^{'}_1/g^{'}_0>c^{'}_l]+c^{'}_l G_0[g^{'}_1/g^{'}_0\geq c^{'}_l]\}=1$$ Question: Is it true that $$Q_0[q_1/q_0<t]\geq Q^{'}_0[q_1/q_0<t]\geq Q^{'}_1[q_1/q_0<t]\geq Q_1[q_1/q_0<t]$$ holds for all $t$?

Any comments about the solution of the question is welcomed. Please also let me know if something is missing or mistaken. Thanks alot.

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$$Q_0[q_1/q_0<t]\geq Q^{'}_0[q_1/q_0<t]\geq Q^{'}_1[q_1/q_0<t]\geq Q_1[q_1/q_0<t]$$
holds. To prove this first I showed that $c_u>c_u^{'}>c_l^{'}>c_l$. Then in the second step I identified that there are altogether $5$ regions due to the relation $c_u>c_u^{'}>c_l^{'}>c_l$ and for each region I simply showed that
$$Q_0[q_1/q_0<t]\geq Q^{'}_0[q_1/q_0<t]\geq Q^{'}_1[q_1/q_0<t]\geq Q_1[q_1/q_0<t]$$
holds. To do this I used the given densities and the $4$ equations which are normalizing the area of the functions $q_i$ and $q_i^{'}$ to $1$.